Formulation of topology optimisation problems with design-dependent loads in the framework of a non-uniform rational basis spline hyper-surfaces

This thesis aims to investigate three different aspect of classic topology optimisation (TO) problems. Firstly, a literature survey on TO methods/algorithms used in literature is conducted. The focus is on both the problem formulation, which is always formulated in a general sense by considering inhomogeneous Neumann-Dirichlet boundary conditions, and the exploitation/recovery of the optimised topology in a computer-aided design (CAD) environment. To this end, particular attention is paid to the TO algorithm based on a pseudo-density field reformulated in the context of non-uniform rational basis spline (NURBS) hyper-surfaces developed at the Institut de Mécanique et d’Ingénierie de Bordeaux by M. Montemurro and collaborators. Unlike classical density-based TO approaches, the NURBS- density-based method separates the pseudo-density field, describing the topology of the continuum, from the mesh of the finite element model. For general TO problem of dimension D, a NURBS entity of dimension D + 1 is used as a topological descriptor. In this way, the topological descriptor relies on a purely geometric entity. In the framework of this approach, the CAD reconstruction phase of the optimised topology becomes a trivial task because the topology boundary is available (at each iteration of the optimisation process) in a CAD-compatible format. Moreover, some fundamental properties of the NURBS basis functions, like the local support property, can be conveniently exploited to determine the gradient of the physical responses with respect to the topological variables, i.e., the pseudo-density evaluated at control points (CPs) and the related weights. Secondly, the main challenges/issues of TO problems with design-dependent loads are addressed. These problems are relevant to those applications involving inertial forces (e.g., gravity, centrifugal forces, etc.) whose intensity depends on the actual topology of the continuum. A new formulation of the penalty function of the inertial forces is proposed to overcome the singularity effect related to the zones characterised by low values of the pseudo-density field. Finally, the case of thermomechanical TO problems is considered. These problems belong to the class of TO problems with design-dependent loads. Indeed, the thermal forces associated with the thermal strain field (occurring in the structure as a result of a given temperature field) depend on the actual distribution of the pseudo-density field. However, unlike the case of the internal forces, this dependence is not explicit, but implicit because it depends on the solution of the thermal problem. Therefore, special care should be taken in the choice of the penalty functions of the main tensors involved in thermomechanical analysis, i.e. the thermal conductivity tensor, the thermal expansion coefficient tensor and the elasticity tensor. All problems are formulated in the most general case of inhomogeneous Neumann-Dirichlet boundary conditions and the effectiveness of the proposed approach is shown on 2D and 3D benchmark problems taken from the literature.